Algebraic, transcendental (i.e., involving trigonometric and exponential functions), ordinary differential equations, or partial differential equations... ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

1. Analytical Method – is one that produces either exact or approximate solutions in closed form 2. Components – the elements of a vector 3. Conformable – matrices with identical dimensions 4. Accuracy – is a measure of the nearness of a value for the true value 5. Precision – is a measure of the clustering of values near each other

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 6. Triangular matrix – a square matrix in which all the elements on one side of the diagonal are zero 7. Gauss elimination – methods for solving a system; reducing the matrix to the upper triangular form, and then back to substitution 8. Double sequence – is a function of domain of ordered pairs (i, j)of integer and with range consisting of a portion of the real number system 9. Non- singular matrix – a square matrix with a non- zero determinant

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology :::: : Where x j (j=1,2,…m) denotes the unknown variable a ij (i=1,2,…n; j=1,2,…m) denotes the coefficients of the unknown variable b i (i=1,2,…n) denotes the non-homogeneous terms

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology  One that produces either exact or approximate solutions in closed form.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. A unique solution – a consistent set of solutions 2. No solution – an inconsistent set of equations 3. An infinite number of solutions – a redundant set of equations 4. The trival solution x j = 0 – a set of homogenous equations

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. Direct methods – are systematic procedures, based on algebraic elimination, that obtain the solution in a fixed number of operations. 2. Iterative methods – obtain the solution asymptotically by an iterative procedure. A trial solution is assumed, the trial solution is substituted into the system of equations to determine the mismatch in the trial solution, and an improved solution is obtained from the mismatch data.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 1 An (m x n) or (m, n) matrix is a rectangular array of quantities arranged in m rows and n columns.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 2 A matrix with only one row is a special kind of matrix known as a row vector.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 3 A matrix with only one column is a special kind of matrix known as a column vector.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 4 The (n x m) or (n, m) matrix obtained from a given (m x n) or (m, n) A by interchanging its rows and columns is called the transpose of A denoted by the symbol A T.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 5 A square matrix is a matrix where the dimensions m is equal to n.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 6 A symmetric matrix is one where a ij = a ji for all i’s and j’s.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 7 A square matrix in which each element not on the principal diagonal is zero is called a diagonal matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 8 A square matrix in which every element below the principal diagonal is zero is said to be upper triangular matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 9 A square matrix in which every element above the principal diagonal is zero is said to be lower triangular matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 10 A square matrix in which all elements equal to zero, with the exception of a band centered on the main diagonal is called a bonded matrix (e.g. tridiagonal matrix).

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 11 A diagonal matrix in which each diagonal element is 1 is called a unit matrix or identity matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 12 A matrix in which every element is zero is called a null matrix or zero matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 13 The determinant of an (n, n) square matrix A is written as lAl and is defined by either of or in which c ij is known as the cofactor of the element a ij.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 14 The cofactor c ij of an (n, n) square matrix A is obtained by first removing row i and column j to form an (n-1, n-1) matrix and then performing the operation

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 15 The augmented matrix is obtained by adjoining the column vector b to the coefficient matrix A.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Definition 16 A coefficient matrix with a zero determinant is singular, a unique solution for x requires a non- singular coefficient matrix.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology A. Methods for Triangular Matrices It involves reduction of matrix equation into one of the forms:, L = lower triangular matrix, U = upper triangular matrix

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology B. Cramer’s Rule  Gives the components x i of x in terms of determinants according to:

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology B. Cramer’s Rule Example  Use the Cramer’s rule to solve: 0.3x x 2 + x 3 = x 1 + x x 3 = x x x 3 = -0.44

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology C. Gaussian Elimination  a method for solving a system of the type (A x = b) wherein the goal is to reduce it to the upper triangular form and then use the back substitution scheme to obtain the components from each of the remaining equations.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology C. Gaussian Elimination Example Use Gaussian elimination to solve: 3x x x 3 = x 1 + 7x x 3 = x 1 – 0.2x x 3 = 71.4

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology D. Gauss-Jordan Method  a variation of Gauss Elimination wherein the goal is to reduce the original matrix to a diagonal form.  not popular since there is neither a reduction in programming complexity nor increased efficiency.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology D. Gauss-Jordan Example Use Gauss-Jordan to solve the previous problem: 3x x x 3 = x 1 + 7x x 3 = x 1 – 0.2x x 3 = 71.4

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology E. LU Decomposition Method  is another elimination method of solving general systems of linear algebraic equations wherein the objective is to find a lower triangular factor L and an upper triangular factor U such that the system of equations can be transformed according to Where A* = matrix after row exchange have been made to allow the factors L and U to be computed accurately; b* = vector b after an identical set of row exchanges.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology E. LU Decomposition Example Solve the previous problem: 3x x x 3 = x 1 + 7x x 3 = x 1 – 0.2x x 3 = 71.4

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. Division by Zero 2. Round-off Errors 3. Ill-Condition Systems  is one where a small changes in one or more of the coefficients results in large changes in the solution. 4. Singular Systems  is worse than ill-conditioned because two equations in the system are identical.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. Pivoting 2. Use of more significant figures 3. Scaling

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1.Solve the following systems: x 1 + 2x 2 = x 1 + 2x 2 = 10.4 Then solve it again, but with the coefficient of x 1 in the second equation modified slightly to 1.05.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 2. Evaluate the determinant of the following systems: 3x 1 + 2x 2 = 18 -x 1 + 2x 2 = 2 And solve also the determinant in prob. 1.

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology A. Gauss-Seidel Method  Iterative or approximate methods.  Start the process by assigning initial values (guessing a value) and then use a systematic method to obtain a refined estimate of the root. Then solve for the subsequent values of x 1, x 2, x 3, etc., using the following equations:

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology A. Gauss-Seidel Method

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology A. Gauss-Seidel Example Solve the previous problem: 3x x x 3 = x 1 + 7x x 3 = x 1 – 0.2x x 3 = 71.4

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Convergence criterion for the Gauss-Seidel and